# Journal of Operator Theory

Volume 84, Issue 2, Fall 2020 pp. 289-322.

$\mathrm{E}$-semigroups over closed convex cones

**Authors**:
Anbu Arjunan (1), R. Srinivasan (2), and S. Sundar (3)

**Author institution:** (1) Institute of Mathematical Sciences (HBNI), CIT Campus, Taramani, Chennai, Tamilnadu, 600113, India

(2) Chennai Mathematical Institute, H1 Sipcot IT Park, Siruseri, Kelambakkam, Tamilnadu, 603103, India

(3)Institute of Mathematical Sciences (HBNI), CIT Campus, Taramani, Chennai, Tamilnadu, 600113, India

**Summary: **In this paper, we study $\mathrm{E}$-semigroups over convex cones. We prove a structure theorem for $\mathrm{E}$-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, $\mathrm{E}_0$-semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of $2$-parameter CCR flows, containing mutually non-cocycle-conjugate $\mathrm{E}_0$-semigroups.

**DOI: **http://dx.doi.org/10.7900/jot.2018sep17.2271

**Keywords: **$\mathrm{E}_0$-semigroups, convex cones, CCR flows

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